Copyright (c)JCPDS-International Centre for Diffraction Data 2002, Advances in X-ray Analysis, Volume 45.
IMPROVING THE ACCURACY OF RIETVELD-DERIVED LATTICE
PARAMETERS BY AN ORDER OF MAGNITUDE
B. H. O’Connor and S. Pratapa
Materials Research Group, Department of Applied Physics, Curtin University of Technology,
GPO Box U1987, Perth, WA 6845, Australia
ABSTRACT
This is the fourth in a series of papers on Rietveld materials characterisation metrology presented
at recent Denver X-ray Conferences [refs 1-3]. Reference 3 examined the management of the
zero-point and specimen displacement corrections in extracting lattice parameters from BraggBrentano x-ray diffraction data by Rietveld analysis. This paper emphasises the importance of
pre-determining the zero-point of the 2θ-scale, 2θ o , prior to, and independently of, the Rietveld
calculations. In addition, the specimen displacement correction must be refined in the Rietveld
computations with Bragg-Brentano data to minimise systematic errors in the lattice parameters.
A zero-point determination procedure is described using line-position standard reference
materials (SRMs) to extrapolate a plot of 2θ bias versus cos θ obs to θ = 90°, with 2θbias = 2θtrue 2θobs for 2θtrue calculated with the known lattice parameters and 2θobs being the measured value.
Excellent agreement was obtained for 2θ o values determined with NIST SRM 660a LaB6 and
NIST Si 640 standards. The effectiveness of this approach when used with the co-refinement of
the lattice parameters and specimen displacement refinement is demonstrated with data sets for
CeO2, Al2O3 and TiO2 (rutile) powders from the NIST SRM 674a set. The accuracy of lattice
parameters determined by this Rietveld approach is approximately 1 part in 50,000.
INTRODUCTION
The Curtin University Materials Research Group has gained extensive experience in the use of
Rietveld analysis of advanced ceramics characterisation with particular reference to phase
composition, linear strain (lattice parameter changes), non-linear strain and crystallite size (line
broadening) and texture. The present study is the latest in a series of Advances in X-ray Analysis
publications on Rietveld analysis metrology [1-3]. This latest study was conducted in view of
their experience in various studies of linear strain by Rietveld analysis with Bragg-Brentano
XRD data that the lattice parameters had poor reproducibility. This experience is consistent with
the IUCr Rietveld round robin study on m-ZrO2 data which found that the accuracy of lattice
parameters was x16 worse than the esd estimates from Rietveld refinements [4].
Reference 3 considers the impact on Rietveld lattice parameter determinations of the heavy
Rietveld parameter correlations between the lattice parameters, the specimen displacement (SD),
and the instrument zero-point of the 2θ measurement scale ( 2θ o ). These correlations cause the
propagation of bias between these parameters.
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159
DETERMINING THE ZERO-POINT INDEPENDENTLY OF THE RIETVELD
CALCULATIONS USING SRMs
The zero-point, 2θ o , for a single Bragg peak is one of the systematic errors contributing to the
quantity θ bias in:
λ = 2d sin( θ obs + θ bias )
(1)
where θ obs is the measured Bragg angle. If θ bias is dominated by the specimen displacement and
zero-point errors, the peaks are biased by:
2θ bias = 2θ o + 2 [ ∆ R / R ] cos θ
(2)
for specimen displacement ∆ R and goniometer radius R .
The zero-point may be determined as follows: (1) measure the 2θ obs using a line-position
standard reference material (SRM), with high-angle data only being acquired; (2) for each of
these reflections, the bias is determined as 2θbias = 2θtrue - 2θobs, where the values 2θtrue are
calculated with the certified lattice parameters for the SRM; and then (3) a plot of 2θbias versus
cos θ obs is prepared and the line-of-best-fit extrapolated to θ obs = 90°. The intercept on the 2θbias
axis gives 2θ o .
It is noted that specimen transparency causes an additional SD contribution according to the
information depth for symmetric optics,
(3)
ID = sin θ / 2µ
where µ is the linear attenuation coefficient. This effect should be included in the extrapolation
procedure and the Rietveld algorithm when working with x-ray transparent materials. In that
case, equation 2 becomes
2θ bias = 2θ o + 2 [( ∆ R + sin θ / 2µ ) / R ] cos θ
(4)
EXPERIMENTAL
The materials employed for the study were (i) NIST SRM 660a LaB6 and SRM 640 Si powders
for constructing 2θ o calibrations, and (ii) the set of NIST SRM 674a powders for CeO2, α-Al2O3
and TiO2 (rutile) for acquiring whole-pattern data for the Rietveld determinations of lattice
parameters.
The XRD data were measured with a Siemens D500 Bragg-Brentano diffractometer configured
as follows - Cu tube [type Fk60-04 CU] operating at 40 kV and 30 mA (Kα wavelengths:
1.54060, 1.54439 Å); fixed slit optics with incident beam divergence = 1°, receiving slit =
0.15°, post-diffraction graphite analyser; and NaI detector with pulse discrimination. In
collecting data sets, the 2θ step size was 0.02°; the counting time per step was 2s; and the 2θ
range = 20 - 150°.
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Copyright (c)JCPDS-International Centre for Diffraction Data 2002, Advances in X-ray Analysis, Volume 45.
The data sets were collected with a 40-position sample holder. The influence of re-mounting the
powder in the holder and also of changing the position of the sample were also investigated.
Rietveld modelling was performed using the Rietica program [5]. Crystal structure data (atom
coordinates) were fixed at values from the Inorganic Crystal Structure Data Base
(FachInformationsZentrum and Gmelin Institut, Germany) - ICSD 72155 from reference [6] for
CeO2, ICSD 73724 for α-Al2O3 [7] and ICSD 64987 for TiO2 (rutile) [8]. The refinements
involved adjusting the pattern-background polynomial parameters (4 terms), the SD, the lattice
parameters, the phase scale and peak profile functions (pseudo-Voigt with asymmetry). The
values of 2θ o and SD were fixed in some refinements and relaxed in others. Transparency
corrections were not applied (see following section).
ZERO-POINT CALIBRATION RESULTS
Figure 1 shows the plots for determining 2θ o by extrapolation. The excellent linearity is
indicated by the regression R2 values. The agreement between the two independent
determinations of 2θ o , 0.0410° for LaB6 and 0.0418° for Si, respectively, is sound. The
difference in the slopes of the 2 powders corresponds to differences in SD, with the effective SD
values being 236 µm for LaB6 and 197 µm for Si, respectively (goniometer radius R = 200.5
mm). It is noted that the corresponding IDs for x-ray attenuation at 2θ o = 0°, 5 µm for LaB6 and
36 µm for Si, are substantially less than the values derived by extrapolation, indicating that
transparency is not dominant.
Based on the 2 calibrations, the value of 2θ o was set at the mean value for the Rietveld
refinements, 0.0414° ± 0.0004°.
RIETVELD REFINMENT TRIALS FOR LATTICE PARAMETERS
Figure 2a shows the CeO2 lattice parameter Rietveld refinement results obtained by fixing the SD
at zero and refining 2θ o . The figure shows the influence of (i) repeating the data collection
without removing the specimen from the holder, (ii) re-packing the powder into the same holder
before recollecting the data, and (iii) recollecting the data by moving the specimen holder to
another position for the 40-sample holder.
The serious mis-match between the values obtained here and the NIST certificate value is
evident. The systematic difference between the values obtained and the NIST value is
approximately 0.003Å or 1 in 2000. Clearly the high correlation between 2θ o and the lattice
parameter forces the latter to a biased value if the SD is not refined. Figure 2b shows the
dramatic improvement realised when 2θ o value is fixed at the calibration value and the SD is
refined. By inspection, the Rietveld results now agree with the NIST value within the stated
limits of uncertainty. It is evident that the true accuracy of the Rietveld lattice parameters is
approximately 0.0005 Å or 1 part in 50,000.
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Copyright (c)JCPDS-International Centre for Diffraction Data 2002, Advances in X-ray Analysis, Volume 45.
Figure 1. Calibration plots to determine the 2θ o value with SRM data. The error bars
indicate ± 1 standard deviation.
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Copyright (c)JCPDS-International Centre for Diffraction Data 2002, Advances in X-ray Analysis, Volume 45.
Figure 2. CeO2 lattice parameters obtained with the 2θ o value (a - above) not fixed at
the calibration value and (b - below) fixed at the calibration value. The error
bars indicate ± 1 standard deviation.
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Copyright (c)JCPDS-International Centre for Diffraction Data 2002, Advances in X-ray Analysis, Volume 45.
Table 1. Analysis of Rietveld precision in the lattice parameters for CeO2 repeat analyses with
data collected after re-packing the powder into the holder. The 2θ o value was fixed at
the calibration value, 0.0414 °, and the SD was refined.
Refinement
1
2
3
4
5
6
7
8
Lattice parameter
(Å)
5.410991
5.410929
5.410916
5.410987
5.410981
5.410975
5.411004
5.410978
Rietveld esd
(Å)
0.000043
0.000042
0.000043
0.000043
0.000048
0.000049
0.000045
0.000045
Mean values
5.410970
0.00045
Esd (pop stats)*
0.000031
(
2
* Esd ( pop stats ) = ∑ x − x)
( n − 1)
)
Table 1 provides a comparison of a population statistics analysis of the uncertainty in CeO2
lattice parameter with the estimated standard deviations provided by the Rietveld program. The
Rietveld repeat values were obtained by fixing the 2θ o at 0.0414º. The esd from population
statistics, 0.000031 Å, is comparable with, but less than the average value from the Rietveld
program which is expected. These results indicate that the accuracy of determining the CeO2
lattice parameter by this method is better than 1 part in 50,000, taking the accuracy to be ± 2
standard deviations as given by the Rietveld program.
Table 2 confirms the accuracy estimate of approximately 1 part in 50,000 from an examination of
the Rietveld lattice parameters for the NIST 674a powders for CeO2, Al2O3 and TiO2 (rutile),
again using the 2θ o calibration value of 0.0414º and with the SD refined. The agreement between
the Rietveld values from this study and the NIST certificate value are all less than 3 standard
deviations of the difference. Taking the accuracy to be ± 2 standard deviations, the values for
CeO2 and α-Al2O3 are better than 1 in 50,000 whereas those for rutile are marginally less.
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Table 2. Lattice parameters from the Rietveld refinement trials for the NIST 674a powders,
compared with the NIST certificate values. The final column gives the ‘accuracy’,
defined as the ratio of the parameter and the esd for the difference between the Rietveld
and certificate values.
Accuracy =
parameter/σ
Parameter
CeO2
a
σ
5.410991
0.000043
5.411102
0.000097
1.1
1:130,000
α-Al2O3
a
σ
c
σ
4.759208
0.000026
12.99202
0.00011
4.759397
0.000080
12.99237
0.00022
2.2
1: 180,000
1.4
1: 120,000
a
σ
b
σ
4.593867
0.000051
2.958941
0.000048
4.593939
0.000062
2.958862
0.000063
0.9
1: 74,000
-1.0
1: 90,000
TiO2,
rutile
This Study
NIST
∆/σ
Certificate
SRM
CONCLUSION
The authors recommend the following strategies for Rietveld lattice parameter determinations
with Bragg-Brentano diffractometer data:
• Determining the zero-point ( 2θ o ) independently of the Rietveld analysis, using SRM line
position standards. The SRM should have minimal x-ray transparency.
• The specimen displacement error must be included in the Rietveld refinement model when
extracting lattice parameters, and the zero-point should be constrained at the calibration
value.
• Only small displacements should modelled (ca. several hundred µm).
Accuracy in the vicinity of 1:50,000 is achievable. Co-refinement of the lattice parameters with
2θo, and no SD refinement, is likely to result in an accuracy of 1 part in several thousand only.
ACKNOWLEDGEMENTS
The authors thank Dr Brett Hunter of the Australian Nuclear Science and Technology
Organisation for assistance with Rietica, and for the provision of SRM 674a materials.
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Copyright (c)JCPDS-International Centre for Diffraction Data 2002, Advances in X-ray Analysis, Volume 45.
S.P. acknowledges (i) the Australian Agency for International Development (AusAID) for
awarding a PhD scholarship, and (ii) the Department of Physics, Institute of Technology 10
September (ITS), Surabaya, Indonesia for providing study leave.
REFERENCES
[1] O’Connor, B.H.; Li, D.Y., Advances in X-ray Analysis 2000, 42, 204-211.
[2] O’Connor, B.H.; Li, D.Y., Advances in X-ray Analysis 2000, 43, 305-312.
[3] O’Connor, B.H.; Li, D.Y; Hunter, B.A, Advances in X-ray Analysis 2001, 44. In print.
[4] Hill, R.J.; Cranswick, L.M.D., J Appl Cryst. 1994, 27, 802-844.
[5] Hunter, B.A., International Union of Crystallography, Commission on Powder Diffraction Newsletter, 1998, 20, 21.
[6] Wolcyrz, M.; Kepinski, L., J Solid State Chem. 1992, 92, 409-413.
[7] Maslen, E.N; Streltsov, V.A; Streltsova, N.R; Ishizawa, N; Satow, Y., Acta Cryst. 1993,
B49, 973-980.
[8] Shintani, H.; Sato, S.; Saito, Y., Acta Cryst. 1975, B31, 1981-1982.
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